1. Portfolio Management-Learning Objective
Basic assumptions of Markowitz portfolio theory?
What is meant by risk, and alternative measures of risk?
Expected rate of return for an individual risky asset and a portfolio of assets?
Standard deviation of return for an individual risky asset and portfolio of assets?
Covariance and correlation between rates of return.

2. Portfolio Management-Learning Objective
Portfolio diversification.
What is the risk-return efficient frontier?
What determines which portfolio on the efficient frontier is selected by an individual investor?
Asset Allocation Models.

3. Markowitz Portfolio Theory
Quantifies risk
Derives the expected rate of return for a portfolio of assets and an expected risk measure
Shows that the variance of the rate of return is a meaningful measure of portfolio risk
Derives the formula for computing the variance of a portfolio, showing how to effectively diversify a portfolio

4. Assumptions of Markowitz Portfolio Theory
Investors consider each investment alternative as being presented by a probability distribution of expected returns over some holding period.
2. Investors minimize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth.

5. Assumptions of Markowitz Portfolio Theory
3. Investors estimate the risk of the portfolio on the basis of the variability of expected returns.
4. Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only.

6. Assumptions of Markowitz Portfolio Theory
5. For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected returns, investors prefer less risk to more risk.

7. Markowitz Portfolio Theory
Using these five assumptions, a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.

8. Alternative Measures of Risk
Variance or standard deviation of expected return
Range of returns
Returns below expectations
Semivariance – a measure that only considers deviations below the mean
These measures of risk implicitly assume that investors want to minimize the damage from returns less than some target rate

9. Expected Rates of Return
For an individual asset - sum of the potential returns multiplied with the corresponding probability of the returns
For a portfolio of assets - weighted average of the expected rates of return for the individual investments in the portfolio

10. Computation of Expected Return for an Individual Risky Investment
Exhibit 7.1

11. Computation of the Expected Return for a Portfolio of Risky Assets
Exhibit 7.2

12. Variance (Standard Deviation) of Returns for an Individual Investment

13. Variance (Standard Deviation) of Returns for an Individual Investment
Exhibit 7.3
Variance ( 2) = .0050
Standard Deviation ( ) =

14. Variance (Standard Deviation) of Returns for a Portfolio
Exhibit 7.4
Computation of Monthly Rates of Return

15. Covariance of Returns
A measure of the degree to which two variables “move together” relative to their individual mean values over time
For two assets, i and j, the covariance of rates of return is defined as:
Covij = E{[Ri - E(Ri)][Rj - E(Rj)]}

16. Covariance and Correlation
The correlation coefficient is obtained by standardizing (dividing) the covariance by the product of the individual standard deviations

17. Covariance and Correlation
Correlation coefficient varies from -1 to +1

18. Correlation Coefficient
It can vary only in the range +1 to -1. A value of +1 would indicate perfect positive correlation. This means that returns for the two assets move together in a completely linear manner. A value of –1 would indicate perfect correlation. This means that the returns for two assets have the same percentage movement, but in opposite directions

19. Portfolio Standard Deviation Formula

20. Portfolio Standard Deviation Calculation
Any asset of a portfolio may be described by two characteristics:
The expected rate of return
The expected standard deviations of returns
The correlation, measured by covariance, affects the portfolio standard deviation
Low correlation reduces portfolio risk while not affecting the expected return

21. Combining Stocks with Different Returns and Risk
Case Correlation Coefficient Covariance a b c d e

22. Combining Stocks with Different Returns and Risk
Assets may differ in expected rates of return and individual standard deviations
Negative correlation reduces portfolio risk
Combining two assets with -1.0 correlation reduces the portfolio standard deviation to zero only when individual standard deviations are equal

23. Constant Correlation with Changing Weights
rij = 0.00

24. Constant Correlation with Changing Weights

25. Constant Correlation with Changing Weights
rij = 0.00

26. Constant Correlation with Changing Weights

27. Portfolio Risk-Return Plots for Different Weights2
With two perfectly correlated assets, it is only possible to create a two asset portfolio with risk-return along a line between either single asset
Rij = +1.00 1
Standard Deviation of Return

28. Portfolio Risk-Return Plots for Different Weights
With negatively correlated assets it is possible to create a two asset portfolio with much lower risk than either single asset
Rij = -0.50 f 2 g h i j
Rij = +1.00 k
Rij = +0.50 1
Rij = 0.00
Standard Deviation of Return

29. Portfolio Risk-Return Plots for Different Weights
Exhibit 7.13
E(R)
Rij = -0.50 f
Rij = -1.00 2 g h i j
Rij = +1.00 k
Rij = +0.50 1
Rij = 0.00
With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no risk
Standard Deviation of Return

30. Estimation Issues
Results of portfolio allocation depend on accurate statistical inputs
Estimates of
Expected returns
Standard deviation
Correlation coefficient
Among entire set of assets
With 100 assets, 4,950 correlation estimates
Estimation risk refers to potential errors

31. Estimation Issues
With assumption that stock returns can be described by a single market model, the number of correlations required reduces to the number of assets
Single index market model:
bi = the slope coefficient that relates the returns for security i to the returns for the aggregate stock market
Rm = the returns for the aggregate stock market

32. The Efficient Frontier
The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return
Frontier will be portfolios of investments rather than individual securities
Exceptions being the asset with the highest return and the asset with the lowest risk

33. Efficient Frontier for Alternative Portfolios
Exhibit 7.15
Efficient Frontier B
E(R) A C
Standard Deviation of Return

34. The Efficient Frontier and Investor Utility
An individual investor’s utility curve specifies the trade-offs he is willing to make between expected return and risk
The slope of the efficient frontier curve decreases steadily as you move upward
These two interactions will determine the particular portfolio selected by an individual investor

35. The Efficient Frontier and Investor Utility
The optimal portfolio has the highest utility for a given investor
It lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility

36. Selecting an Optimal Risky Portfolio
Exhibit 7.16
U3’
U2’
U1’ Y U3 X U2 U1